Which statement about a system of point charges that are fixed in space is necessarily true? Assuming electrostatic potential energy at infinity to be zero.

Options

A. If the electrostatic potential energy of the system is negative, net positive work by an external agent is required to take the charges in the system back to infinity.
B. If the electrostatic potential energy of the system is zero, all charges in the configuration cannot have same sign.
C. If the electrostatic potential energy of the system is negative, net positive work by an external agent was required to assemble the system of charges.
D. If the electrostatic potential energy of the system is negative, then there is no electric force anywhere in space on any other charged particle not part of the system.

Answer

To solve the question regarding the system of point charges that are fixed in space, we need to analyze the properties of electrostatic potential energy and the implications of its values.

### Step-by-Step Solution:

1. **Understanding Electrostatic Potential Energy (U)**:
The electrostatic potential energy of a system of point charges is defined as the work done to assemble the charges from infinity to their respective positions. The formula for the potential energy between two point charges \( q_1 \) and \( q_2 \) separated by a distance \( r \) is given by:
\[ U = \frac{k \cdot q_1 \cdot q_2}{r} \]
where \( k \) is Coulomb's constant.

2. **Potential Energy for Multiple Charges**:
For a system of multiple charges, the total potential energy \( U \) can be calculated by summing the potential energies of all unique pairs of charges:
\[ U = \frac{1}{2} \sum_{i=1}^{n} \sum_{j=1, j \neq i}^{n} \frac{k \cdot q_i \cdot q_j}{r_{ij}} \]
Here, \( r_{ij} \) is the distance between charges \( q_i \) and \( q_j \).

3. **Analyzing the Sign of Potential Energy**:
- If the potential energy \( U \) is **negative**, it indicates that work must be done against the attractive forces to separate the charges to infinity. Thus, positive work is required to disassemble the system.
- If \( U \) is **zero**, it implies that the system is in a neutral state where the forces are balanced, and no work is required to move the charges to infinity.

4. **Evaluating the Statements**:
- **Statement 1**: "If the electrostatic potential energy of a system is negative, then net positive work by an external agent is required to take the charges in the system back to infinity." This statement is **true**.
- **Statement 2**: "If the electrostatic potential energy of a system is zero, then all the charges in the configuration cannot have the same sign." This statement is also **true** because if all charges had the same sign, the potential energy would be positive.
- **Statement 3**: "If the electrostatic potential energy is negative, then net positive work was required to assemble the system of charges." This statement is **false**; negative potential energy indicates that work was done by the system to assemble the charges.

5. **Conclusion**:
From the analysis, we conclude that statements 1 and 2 are necessarily true regarding the system of point charges.

### Final Answer:
The statements that are necessarily true are:
1. If the electrostatic potential energy of a system is negative, then net positive work by an external agent is required to take the charges in the system back to infinity.
2. If the electrostatic potential energy of a system is zero, then all the charges in the configuration cannot have the same sign.